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III. Note on Foucault's Gyroscope. By M. J. BERTRAND*.
THE
HE ingenious apparatus to which M. Léon Foucault has

given the name of gyroscope, is now well known to all philosophers. There is no necessity, therefore, for me here to redescribe the same; it will be sufficient to recall to mind that, essentially, it consists of a solid of revolution turning rapidly around its axis of figure at the same time that the latter is obliged, by the nature of the system, to remain in a plane fixed with respect to the earth. The movement of the axis of the gyroscope in this plane, however, is perfectly free.

The explanation, almost intuitive, of the observed phænomena is to be sought in the principles discovered by Poinsot, and the following note is but a corollary to the admirable memoir written by this celebrated geometer twenty years ago, and published entire in the 16th volume of the first series of Liouville's Journal.

It is well known that Poinsot regards each molecule of a moving body as animated by a force equal to the product of its mass into its velocity. All the forces which at a given moment animate the molecules of a moving solid body, may be composed by the rules of statics, and reduced to a force and a couple; if the solid body is free, and not solicited by any exterior force, this resultant force and couple are invariable. But if the influence of exterior forces is superadded to that of inertia, the system of forces which animate the body at the expiration of an infinitely small interval of time dt, may be considered as composed of two others; first, the system of finite forces which animated the body at the commencement of the interval under consideration ; and secondly, the system of exterior forces which have acted on the body, each multiplied by the magnitude dt of that interval.

This fundamental principle conducted Poinsot to his most elegant theorems, and as will be seen, it suffices for the complete explanation of the phænomena discovered by Foucault.

We shall suppose the apparatus to be so disposed that the axis of rotation, which is the axis of symmetry of the rotating body, is compelled to remain in a plane P, fixed in relation to the earth. Let o be the centre of the instrument, conceived as fixed, and let us examine solely the motion of the system around this point, reducing, consequently, all the forces to the couples which they produce.

Let oА be the actual position of the axis of rotation in the plane P, and ol the parallel to the earth's axis drawn through the point o.

* From the Journal de Mathématiques pures et appliqués, 2nd series, vol. i. p. 379.

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In order that the axis oA may remain apparently at rest in the plane P, it must in reality turn round ol with a velocity equal to that of the earth, and in twenty-four hours describe a cone of revolution. Let oA' be a position upon this cone infinitely close to oA. The couple which animates the gyroscope turning, during the first instant, around 0A, has its axis directed along this line, and equal to the product of the moment of inertia into the angular velocity w. In order that this axis, which we will represent by og, may, during the following instant, become oG (directed along A'), the system, during the infinitely small period dt, must have been solicited by a couple directed along GG', and having an intensity represented by

GG

dt Now the only action directly experienced by the instrument is the reaction of the fixed plane P; this reaction can only produce forces perpendicular to the plane, and, consequently, a couple with its axis situated in this plane. The line GG', therefore, must be parallel to the plane P, and for this reason P must be a tangent plane to the cone, and hence perpendicular to the plane We have then this first theorem :

The axis of the gyroscope being compelled to remain in a plane P, it cannot remain in equilibrium unless it coincides with the projection of the parallel to the earth's axis upon the fixed plane.

When this coincidence does not exist at the commencement, relative equilibrium is impossible, and the instrument is set in oscillation, the laws of which we must calculate.

We may at once remark, that, whatever may be the initial position oÀ of the axis, it is allowable to apply to the instrument two equal and contrary couples, one of which would, alone, retain the axis in a state of apparent repose without changing the velocity of rotation. Now, according to the demonstration of the preceding theorem, the axis of this couple is perpendicular to the plane IoA, and its moment is easily seen to be

uww, sin IA; where

M

is the moment of inertia of the gyroscope, w the velocity of the earth’s rotation, and w, the angular velocity of the instrument. But, inasmuch as this couple maintains the axis of the gyroscope in apparent repose, it is the equal and contrary couple which causes the instrument to move.

The latter is decomposable into two others: the one whose xis is in the plane P will be destroyed; the other, having its axis perpendicular to P, is alone efficacious, and is represented by

uww, sin IoA cos (R, IOA);

.

where (P, IOA) designates the dihedral angle formed by the plane P and the plane IoA. Now in the solid angle formed by the three planes ', IOA, and loh, perpendicular to P, we have

sin IoA. cos (P, TOA)=cos IoH .sin AoH; and as the angle IoH is constant, we see that the accelerating couple is proportional to the sine of the deflection of the axis from its position of equilibrium. Hence the law of the oscillations is that of the simple pendulum, and their duration is proportional to the square root of the cosine of the angle formed by the earth's axis and the plane P.

Such is the simple explanation of the observed phænomena. I ought to observe, however, that after having found the expression for the couple which moves the instrument, it is still necessary to explain why the velocity acquired tends to sustain itself ; for, properly speaking, there is here no inertia as in the case of a material point. We know, in fact, that the axis oA having a motion in the fixed planę, the instrument does not turn precisely around 0A, but around an axis making a small angle with the same, and situated in a plane through oA perpendicular to P. This axis not being a principal axis of inertia, tends to displace itself, and describe a small cone; but in order to describe this cone, it would be necessary for it to traverse the fixed plane, which, on its part, resists and produces a couple, whose effect is to raise the axis again, and to sustain, purely and simply, its velocity tangential to the plane P, which latter is increased by the accelerating couple above calculated.

I may add, lastly, that the small angle formed by the axis of the gyroscope, and the veritable axis of rotation having been neglected, the formulas found are but approximative, and consequently do not coincide with the vigorous results obtained by the elaborate, but much more difficult method of Bouz.

IV. On the Anticlinal Line of the London and Hampshire

Basins. By P.J. MARTIN, Esq.

(Continued from vol. xii. p. 452.] BE EFORE we leave the great expanse of tertiary drift between

the South Downs and the Sussex coast, it will be well to turn our attention again to the nature of its principal ingredient, and to consider the great significance of its presence there. Loam is not an original production, nor is it a common alluvium like the mud silt or sediment of quiet waters, ancient or modern. It is a mixture of sand and clay produced by water in brisk agitation; and when clay predominates in the mixture, it is the “brick-earth” of geologists. Neither in the form of loam in Phil. Mag. S. 4. Vol. 13. No. 83. Jan. 1857.

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its more sandy state, or mixed up more or less with angular flints, or as brick-earth clay, has it any of the characteristics of a quiet deposit, - littoral, marine, or lacustrine. It has seldom even the rude multitudinous stratification of more sandy drift. It does not contain any organic remains proper to itself, but now and then only a few broken shells, or perhaps occasionally an entire one derived from the materials of the stratified beds from which it was formed; it is, as I say, neither of marine, nor lacustrine, nor estuary origin, but essentially diluvial.

I have already spoken of the manufacture of loảm. The expression is not misapplied, for this sort of drift is the handiwork of a powerful and active agency. The question then naturally arises, of what was the source of the materials of so many loams and beds of brick-earth in the district we are reviewing? The answer is not difficult. The materials of the stratified tertiaries once covering the outlying chalk, to say nothing of those which were spread over the chalk of the adjoining downs, were exactly the materials fitted to form these loams and brick-clays. The segregation of the latter here and there into beds of a choicer kind, and the admixture of fewer flints, as may be seen near Shopwick and Hampnet at Siddlesbam, and at Fishbourne brick-yards, supposes nothing more than the existence of breaks in the banks of Aint and gravel, or of depressions in the subjacent stratified beds, excavated like the hollows in the sand countries below the chalk, filled up with the slush and turbid waters of the floods, as the sand hollows have been with sand and rubble.

Boulder drift.-Returning again to the coast-line at Selsey, we have yet to assign its proper place to this deposit. For the present it will be best to speak of it as part of a zone external to that which I have described as tertiary or supracretaceous* The corresponding parts of this zone are to be found in the valley of the Thamest; connected with this on the south coast, or in what may be called the Valley of the Solent, by fragmentary remains which may yet be found with the relics of the tertiaries on the Hampshire and Wiltshire chalk. Some of the blocks of stone at Stonehenge are not properly greywethers sandstones; they are crystalline, and could never have been derived from

any other source than the erratics of this zone. Professor Ramsay informs me that the inner circles of stone at Stonehenge are of greenstone, and the altar stone is a felspathic trap. This valuable information is corroborated by reference to an interesting paper from the pen of the Rev. W. D. Conybeare, published in the Gentleman's Magazine for November 1833. Mr. Conybeare does not speak of the altar stone. And there being no better way * Phil. Mag. vol. ii. S. 4.

p.

285.
† Lyell's 'Manual of Geology,' p. 132, 5th edit.

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of accounting for the presence of stones so foreign to the country as these obelisks of greenstone, he falls into the humour of the archæologists and believes them to have been transported from Ireland* The historians and ethnologists who have better studied, in its broad extent, the prevalence of stone-pillar worship in the rude times of Abury and Stonehenge, will be better able to weigh the evidence of the means of transport enjoyed by the Britons of those days, than the zealous men who gave their assistance to the Stukeleys, and Hoares and Cunningtons of the past century. For a knowledge of the above-mentioned paper of Mr. Conybeare I am also indebted to Professor Ramsay.

I have little doubt that an attentive examination of what have hitherto passed as greywethers on the Berkshire, Hampshire, and Wiltshire Hills, and the many which are used for fences and landmarks and other ceconomic purposes in the Vale of Pewsey and other parts of the chalk country, would bring to light many other testimonials of the former existence there of the “ northern drift.” The same may be said, perhaps, of the Hampshire and Dorset coasts, and of the Chesil Bank. The manner in which the occurrence of this zone of drift bears on the general question of dislocation and denudation will be seen presently.

We are now in a position to understand the true geological relations of the country we have had under review, and to consider it as a sectional part of the great anticlinal to which it properly belongs. We see in it, in conjunction with other parts of the same area, fragmentary records of three great geological epochs posterior to the chalk.

The first involves considerations of the deposit and area of the tertiary formations.

The second the “glacial period.” Both these are anterior to the rise of the great anticlinal.

The third commences with this upburst of the older strata, and is the diluvial epoch :-in which I find the parentage of that heterogeneous and much-abused mass into which everyone throws the materials which he cannot otherwise conveniently dispose of—the post-pliocene, and the great undefined universalityDrift.

Tertiary Epoch and Area. When I began to speculate on these matters, one of my first and earliest efforts was directed to disabuse the public mind of the notion, adopted from the French geologists, of the formation

* Mr. Conybeare had not then heard of a “glacial period,” and the transport into this part of the world by icebergs.

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